Mechanical Vibrations (ME-502) - Unit 4 Short Notes

UNIT 4: Mechanical Vibrations – Short Notes

I. Fundamental Concepts

Causes of Vibration

• Unbalanced forces (rotating/reciprocating masses)

• Elasticity of components

• External excitations (periodic, impact, random)

• Friction and backlash

Degree of Freedom (DOF)

• Definition: Minimum number of independent coordinates required to define the system's configuration.

• Examples:

  • 1 DOF: Mass on a spring (vertical translation).

  • 2 DOF: Two masses connected by springs.

  • Multi-DOF: Vehicle suspension (heave, pitch, roll), multi-storey building.

Simple Harmonic Motion (SHM)

• Mathematical definition: \( x = A \sin(\omega t + \phi) \)

  • \(A\): amplitude, \(\omega\): angular frequency, \(\phi\): phase angle.

• Vector representation:

  • Displacement \(x\), velocity \(v = \omega A \cos(\omega t + \phi)\), acceleration \(a = -\omega^2 A \sin(\omega t + \phi)\).

  • All represented as rotating vectors (phasors) with \(\omega\) rad/s.

Energy Methods

• Rayleigh’s principle: For conservative SDOF system, maximum kinetic energy = maximum potential energy.

  • \(\frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} k x_{\text{max}}^2 \implies \omega_n = \sqrt{k/m}\).

  • Used to estimate natural frequency for complex systems by assuming a mode shape.

Equation of Motion (EOM)

• Derivation: Apply Newton’s second law to the mass after drawing Free Body Diagram (FBD).

• General form (SDOF): \( m\ddot{x} + c\dot{x} + kx = F(t) \).

II. SDOF Systems: Free Vibration

Undamped Free Vibration

• EOM: \( m\ddot{x} + kx = 0 \)

• Natural frequency: \( \omega_n = \sqrt{\frac{k}{m}} \) (rad/s), \( f_n = \frac{\omega_n}{2\pi} \) (Hz).

• General solution: \( x(t) = A \sin(\omega_n t) + B \cos(\omega_n t) = C \sin(\omega_n t + \phi) \).

Viscously Damped Systems

• EOM: \( m\ddot{x} + c\dot{x} + kx = 0 \)

• Damping ratio: \( \zeta = \frac{c}{c_c} \), where critical damping constant \( c_c = 2\sqrt{km} \).

• Damped natural frequency: \( \omega_d = \omega_n \sqrt{1 - \zeta^2} \) (only for \(\zeta < 1\)).

• Cases:

  • Underdamped (\(\zeta < 1\)): Oscillatory decay.

  • Critically damped (\(\zeta = 1\)): Fastest return to equilibrium without oscillation.

  • Overdamped (\(\zeta > 1\)): Slow non-oscillatory return.

Logarithmic Decrement

• Definition: \( \delta = \ln \left( \frac{x_1}{x_2} \right) \), where \(x_1, x_2\) are successive peak amplitudes.

• Relation to \(\zeta\): \( \delta = \frac{2\pi\zeta}{\sqrt{1 - \zeta^2}} \) (valid only for \(\zeta < 1\)).

• [!TIP] Cannot be used for \(\zeta \geq 1\) because no successive peaks exist.

Coulomb Damping (Dry Friction)

• Constant friction force \(F_f = \mu N\) opposing motion.

• EOM: \( m\ddot{x} + kx = \pm F_f \) (sign changes at velocity reversal).

• Amplitude decay: Linear reduction per cycle: \( x_n = x_0 - \frac{2nF_f}{k} \).

• Differences from viscous damping:

  1. Amplitude decay is linear (vs. exponential).

  2. Frequency independent of amplitude (vs. slight dependence for viscous).

  3. Friction force constant (vs. velocity-proportional).

  4. Energy loss per cycle constant (vs. proportional to \(v^2\)).

Hysteresis & Structural Damping

• Hysteresis loop: Force vs. displacement plot for cyclic loading; area = energy dissipated per cycle.

• Equivalent viscous damping: \( c_{eq} = \frac{\text{Energy loss per cycle}}{2\pi \omega_n X^2} \).

• Structural damping: Modeled as complex stiffness \( k(1 + i\eta) \), where \(\eta\) is loss factor.

Example Calculations Pattern

1. Given \(m, k, c\): Compute \(c_c, \zeta, \omega_d, \delta\).

2. Effect of spring constant change: \( \omega_n \propto \sqrt{k} \), so \( T \propto 1/\sqrt{k} \).

III. SDOF Systems: Forced Vibration

Harmonic Forcing \( F(t) = F_0 \sin \omega t \)

• Steady-state solution: \( x(t) = X \sin(\omega t - \phi) \)

• Magnification Factor (MF):

  \[ M = \frac{X}{X_{st}} = \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}, \quad r = \frac{\omega}{\omega_n} \]

  where \( X_{st} = F_0/k \) (static deflection).

• Phase angle:

  \[ \phi = \tan^{-1}\left( \frac{2\zeta r}{1 - r^2} \right) \]

• Resonance:

  • For \(\zeta < 1/\sqrt{2}\), \(M_{\text{max}} \approx \frac{1}{2\zeta}\) at \( r \approx \sqrt{1 - 2\zeta^2} \).

  • For \(\zeta \geq 1/\sqrt{2}\), no peak (MF decreases monotonically).

Total Response

• \( x(t) = x_h(t) + x_p(t) \)

  • \(x_h\): Homogeneous (transient) solution (decays with damping).

  • \(x_p\): Particular (steady-state) solution (persists).

• Determined by applying initial conditions \(x(0), \dot{x}(0)\).

Base Excitation

• Base motion: \( y(t) = Y \sin \omega t \)

• Transmissibility:

  \[ T = \frac{X}{Y} = \frac{r^2}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}} \]

• Relative displacement: \( X_r = X - Y = \frac{F_0}{k} \cdot \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}} \) (same as MF form).

Rotating Unbalance

• Equivalent force: \( F_0 = m_e r \omega^2 \), where \(m_e\) = eccentric mass, \(r\) = eccentricity.

• Response: Same as harmonic force with \(F_0\) above.

Work Done in Harmonic Motion

• Over one cycle: \( W_{\text{cycle}} = \pi F_0 X \sin\phi \) (net work = energy dissipated by damping).

• Over first second or first quarter: Compute by integrating \( F(t) \dot{x}(t) \, dt \) over interval.

IV. Applications of SDOF Systems

Gun Recoil Mechanism

• Design goal: Critically damped (\(\zeta = 1\)) to minimize rebound and time to stop.

• Recoil distance \(d\): From energy conservation: \( \frac{1}{2} m v_0^2 = \frac{1}{2} k d^2 + c_c v_0 d \) (for critical damping, \(c_c = 2\sqrt{km}\)).

• Solve for \(k\) given \(m, v_0, d\).

Vehicle Dynamics – Trailer Suspension

• Critical speed \(v_c\): When excitation frequency \(\omega = \omega_n\).

  \[ v_c = \frac{\omega_n L}{2\pi}, \quad \text{where } L = \text{road wavelength} \]

• Amplitude at speed \(v\):

  \[ X = Y \cdot \frac{r^2}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}, \quad r = \frac{v}{v_c} \]

  \(Y\) = road roughness amplitude.

Car Roll Vibrations

• Natural frequency for roll mode:

  \[ \omega_n = \sqrt{\frac{k_h \cdot h}{m \cdot k^2}} \]

  where \(k_h\) = roll stiffness (sum of spring constants × track width²/2), \(h\) = CG height, \(k\) = radius of gyration about roll axis.

Machine Foundation on Resilient Base

• Given static deflection \(\delta_{st} = \frac{W}{k}\) (where \(W = mg\)).

• \(\omega_n = \sqrt{g / \delta_{st}}\).

• At resonance (\(\omega = \omega_n\)), \(X = Y / (2\zeta)\).

• Dynamic force on base: \( F_d = k (X - Y) \).

• Relative amplitude: \( X_r = X - Y \).

Vibration Isolation

• Goal: Reduce transmissibility \(T\) for \(\omega > \sqrt{2}\omega_n\).

• Effect of damping: Increases \(T\) near resonance but broadens isolation bandwidth.

• Design: Choose \(\omega_n\) low enough so that operating \(\omega \gg \omega_n\), but avoid too low \(\omega_n\) (large static deflection).

• Example: Compressor mounting – isolate forcing frequency \(f\) by setting \(f_n < f/\sqrt{2}\).

V. Multi-DOF Systems

Undamped Free Vibration

• EOM: \( \mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{0} \)

• Assume solution \( \mathbf{x} = \boldsymbol{\phi} e^{i\omega t} \).

• Eigenvalue problem: \( (\mathbf{K} - \omega^2 \mathbf{M})\boldsymbol{\phi} = \mathbf{0} \).

• Characteristic equation: \( \det(\mathbf{K} - \omega^2 \mathbf{M}) = 0 \) → gives \(n\) natural frequencies \(\omega_{n1}, \dots, \omega_{nn}\).

• Mode shapes \(\boldsymbol{\phi}^{(r)}\): Non-trivial solutions for each \(\omega_{nr}\).

Orthogonality Properties

• Mass orthogonality: \( \boldsymbol{\phi}^{(i)T} \mathbf{M} \boldsymbol{\phi}^{(j)} = 0 \) for \(i \neq j\).

• Stiffness orthogonality: \( \boldsymbol{\phi}^{(i)T} \mathbf{K} \boldsymbol{\phi}^{(j)} = 0 \) for \(i \neq j\).

• Enables decoupling of equations via modal transformation.

Normal Modes

• Each mode vibrates independently at its natural frequency when excited.

• Node: Point in a mode shape with zero displacement.

• Sketch: For 2-DOF system, first mode (in-phase), second mode (out-of-phase) with one node between masses.

Principal Coordinates

• Transformation: \( \mathbf{x} = \boldsymbol{\Phi} \mathbf{q} \), where \(\boldsymbol{\Phi} = [\boldsymbol{\phi}^{(1)} \dots \boldsymbol{\phi}^{(n)}]\).

• Substituting into EOM and using orthogonality gives decoupled equations:

  \[ \ddot{q}_r + \omega_{nr}^2 q_r = 0 \quad \text{(undamped)} \]

• Example: For 2-DOF system, solve eigenvalue problem to get \(\omega_{n1}, \omega_{n2}, \boldsymbol{\phi}^{(1)}, \boldsymbol{\phi}^{(2)}\), then compute \(q_1, q_2\) from initial conditions.

Example Systems

• Two-mass, three-spring: Write EOM, form \(\mathbf{M}, \mathbf{K}\), solve det equation.

• Torsional systems: Replace \(m\) with polar inertia \(J\), \(k\) with torsional stiffness \(GJ/L\).

• Springs in series/parallel: Reduce to equivalent stiffness before forming matrices.

VI. Special Topics

Whirling of Shafts

• Critical speed: Rotational speed at which shaft deflection becomes large due to resonance with natural transverse frequency.

• Proof: Critical speed \(\omega_c = \omega_n\) (natural frequency of transverse vibration for the shaft-disk system).

• Rotor safety: Operating speed should be < 70% or > 120% of first critical speed to avoid resonance.

Dynamic Vibration Absorber

• Principle: Attach a tuned mass-spring-damper to primary system to reduce vibration at specific frequency.

• Tuning condition (undamped absorber): \( \omega_a = \omega_p \) (absorber natural frequency = excitation frequency).

• Result: Primary system amplitude becomes zero at \(\omega_p\) (if no damping in absorber).

Torsional Vibrations

• Natural frequency for multi-inertia system:

  \[ \omega_n = \sqrt{\frac{k_t}{J}} \]

  where \(k_t = GJ/L\) (torsional stiffness), \(J\) = equivalent inertia.

• Procedure: Write torque equations, form \(\mathbf{J}, \mathbf{K}_t\) matrices, solve \(\det(\mathbf{K}_t - \omega^2 \mathbf{J}) = 0\).

Continuous Systems – Beam Vibration

• Euler-Bernoulli theory: \( EI \frac{\partial^4 y}{\partial x^4} + \rho A \frac{\partial^2 y}{\partial t^2} = 0 \).

• Natural frequency depends on:

  • \(E\) (Young’s modulus), \(I\) (area moment of inertia), \(\rho\) (density), \(A\) (cross-sectional area).

  • Boundary conditions (simply supported, cantilever, etc.).

• Example: Pump on simply supported beam – avoid operating frequency within \(\pm 3\text{ Hz}\) of beam’s natural frequency. Compute \(I\) from \( f_n = \frac{\pi}{2L^2} \sqrt{\frac{EI}{\rho A}} \).

VII. Noise and Acoustics

Sound Pressure Level (SPL)

• Definition: \( L_p = 20 \log_{10}\left( \frac{p}{p_0} \right) \) dB, where \( p_0 = 20\ \mu\text{Pa} \) (threshold of hearing).

• Relationship with sound power level \(L_W\):

  \[ L_W = L_p + 10 \log_{10}\left( \frac{A}{A_0} \right) \]

  where \(A\) = absorbing area, \(A_0 = 1\ \text{m}^2\).

Sound Intensity

• Inverse square law: \( I \propto \frac{1}{r^2} \).

• dB change when distance doubled: \( \Delta L = 10 \log_{10}\left( \frac{I_2}{I_1} \right) = 10 \log_{10}\left( \frac{1}{4} \right) = -6\ \text{dB} \).

Octave Band Analysis

• Purpose: Divide frequency spectrum into bands (center frequencies: 31.5, 63, 125, 250, 500, 1000, 2000, 4000, 8000 Hz) to identify dominant noise frequencies.

• Procedure: Use filters to measure SPL in each band; plot spectrum.

Human Response to Noise

• Effects: Hearing loss, stress, speech interference, annoyance.

• Permissible limits: OSHA: 90 dB(A) for 8 hrs; NIOSH: 85 dB(A) for 8 hrs (3 dB exchange rate).

Hearing Conservation & Damage Risk

• Standards: OSHA (90 dB/8h), NIOSH (85 dB/8h), EU directives.

• Exchange rate: 3 dB (NIOSH) vs. 5 dB (OSHA) – halving exposure time per 3/5 dB increase.

• Exposure duration: \( T = \frac{T_0}{2^{(L - L_0)/\text{exchange}}} \).

Noise Control Precautions

• At source: Balance rotors, use quieter processes, damp vibrations.

• Along path: Barriers, enclosures, absorptive linings.

• At receiver: Ear protection (NRR rating), administrative controls.

VIII. Vibration Measurement and Analysis

Vibrometers & Accelerometers

• Principle: Seismic instrument – mass-spring-damper system with relative displacement measurement.

• Frequency response:

  • Displacement: \( \frac{X}{Y} = \frac{r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \)

  • Velocity: \( \frac{V}{Y} = \frac{r^2 \omega_n}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \)

  • Acceleration: \( \frac{A}{Y} = \frac{r^2 \omega_n^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \)

• Use: Accelerometer for high frequency, vibrometer for low frequency.

Fourier Series for Impact Forces

• Periodic impact force \(F(t)\) with period \(T\) expanded as:

  \[ F(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos n\omega t + b_n \sin n\omega t \right) \]

  where \( \omega = 2\pi/T \), coefficients:

  \[ a_n = \frac{2}{T} \int_0^T F(t) \cos n\omega t \, dt, \quad b_n = \frac{2}{T} \int_0^T F(t) \sin n\omega t \, dt \]

• Example: Forging hammer – rectangular pulse within period.

IX. Advanced Problem Solving

Rayleigh’s Method

• Estimate first natural frequency by equating max KE and max PE with assumed mode shape \(\phi(x)\):

  \[ \omega_n^2 = \frac{\int_0^L EI \left( \frac{d^2\phi}{dx^2} \right)^2 dx}{\int_0^L \rho A \phi^2 dx} \]

• Accuracy depends on closeness of assumed shape to actual first mode.

Logarithmic Decrement Limitations

• Not applicable for \(\zeta \geq 1\) (no oscillations → no successive peaks).

• For \(\zeta \approx 1\), use step response or half-power bandwidth method.

Work-Energy Principles

• For conservative systems: \( T + V = \text{constant} \).

• For damped systems: Work done by damping = loss of mechanical energy.

Hysteresis Energy Loss per Cycle

• \( \Delta E = \pi k \eta X^2 \), where \(\eta\) = loss factor.

Combining Free & Forced Responses

• Total response \(x(t) = x_{\text{transient}}(t) + x_{\text{steady-state}}(t)\).

• Transient decays with time; steady-state persists.

• Use initial conditions to determine constants in transient part.

Key Formulas Summary

[!EXAM TIP]

• SDOF applications (gun recoil, trailer, machine foundation) are numerical hotspots – practice deriving formulas from first principles.

• MDOF: Focus on 2-DOF systems – setting up matrices, solving eigenvalue problem, orthogonality.

• Noise: SPL, octave bands, hearing conservation are short note favorites.

• Whirling & dynamic absorber: Understand definitions and tuning conditions.

• Always check units consistency (N, kg, m, s, rad).